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Non-Isomorphic Cayley Graphs of Metacyclic Groups of Order 8 p with the Same Spectrum

Author

Listed:
  • Lang Tang

    (School of Mathematics and Statistics, Hunan First Normal University, Changsha 410205, China)

  • Weijun Liu

    (School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China
    College of General Education, Guangdong University of Science and Technology, Dongguan 523083, China)

  • Rongrong Lu

    (School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China)

Abstract

The spectrum of a graph Γ , denoted by S p e c ( Γ ) , is the multiset of eigenvalues of its adjacency matrix. A Cayley graph C a y ( G , S ) of a finite group G is called Cay-DS (Cayley graph determined by its spectrum) if, for any other Cayley graph C a y ( G , T ) , S p e c ( C a y ( G , S ) ) = S p e c ( C a y ( G , T ) ) implies C a y ( G , S ) ≅ C a y ( G , T ) . A group G is said to be Cay-DS if all Cayley graphs of G are Cay-DS. An interesting open problem in the area of algebraic graph theory involves characterizing finite Cay-DS groups or constructing non-isomorphic Cayley graphs of a non-Cay-DS group that share the same spectrum. The present paper contributes to parts of this problem of metacyclic groups M 8 p of order 8 p (with center of order 4), where p is an odd prime, in terms of irreducible characters, which are first presented. Then some new families of pairwise non-isomorphic Cayley graph pairs of M 8 p ( p ≥ 5 ) with the same spectrum are found. As a conclusion, this paper concludes that M 8 p is Cay-DS if and only if p = 3 .

Suggested Citation

  • Lang Tang & Weijun Liu & Rongrong Lu, 2025. "Non-Isomorphic Cayley Graphs of Metacyclic Groups of Order 8 p with the Same Spectrum," Mathematics, MDPI, vol. 13(12), pages 1-14, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:12:p:1903-:d:1673366
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